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Okay, so good morning everyone. So in our last lecture we were working on deformation of a

nanorod which is like your one dimensional crystal, one dimensional crystal and it is

subjected to uniform v k. Okay and then we thought what would be that deformation map.

So we took a hint from the continuum deformation map which was that little x of s which is same

as little x of big X 1, big X 2 and s that is equal to little x of f plus e to the power s k

into little x naught minus little x of f plus s tau k hat. Okay and this was the cross section

at any s maybe I should have capital S here and that is the cross section for s equal to 0. So,

you could write down the map for any cross section in terms of the cross section at s

equal to 0 and that was your unknown. So, we are going to get some hint from this continuum map

and say that the nanorod would also follow this map but it is just that instead of cross section

we have the small repeating box of the nanorod or the unit cell of the 1D crystal. Right remember

we drew our unit cell for the nanorod or 1D crystal. So, it can extend in both the directions

if it is an infinite nanorod and then you can think of atoms within the nanorod. Okay,

so that is one unit cell of the nanorod. So, this is going to act like your cross section

and there are so many unit cells, so there are so many cross sections. So, unit cells over here

are the analogues of cross section. Okay and the cross section you are denoting by little x naught

which was only a function of big x 1 big x 2 and the unit cell you denote by little x naught comma

j. Okay, where j denotes how many atoms are present within the unit cell. In case of 3 then j will

simply run from 1 to 3. Okay, if you have m atoms within the unit cell j runs from 1 to m. Okay,

so we then think of what could be the discrete map for the nanorod and we wrote down it would

become little x of i comma j that is equal to the same little x f because x f was defined in terms

of v and k. I guess it was v cross k divided by magnitude of k. You can look back at the formula

which was also the fixed point plus e to the power then s, s was denoting the arc length from s equal

to 0. So, if you are looking at ith unit cell and the length of these unit cell or the period in

this reference state in the straight state if that's l naught, if this is l naught then your s is

simply i times l naught. Isn't it? That's how much you move along the units along the 1d crystal. So,

this becomes e to the power i l naught k and then little x naught is simply your little x naught comma

j minus little x f plus i l naught tau k hat. Okay, you see so this is your discrete map that

we are looking for and now this little x naught comma j you have to understand that this does

not depend on big X 1 big X 2. So, this is there is no big X 1 big X 2 present. In case of continuum

you had the cross section which was spanned in the e 1 e 2 plane. So, you had big X 1 big X 2

coordinates, but over here in case of discrete version there is no big X 1 big X 2 their role

is taken by which quantity over here? j it is the j which takes the role of big X 1 big X 2. So,

as you change your j you know it's like it's going to span your e 1 e 2 plane it's going to

span your cross section. Think of you know a nanorod and this is one of the cross sections.

If it's a continuum then you have big X 1 big X 2 coordinate and you have material point

everywhere for every big X 1 big X 2 you have got a material point, but if you want to think of it

in a discrete way then you don't have material point everywhere you have material point only

at the locations of the atoms right. So, maybe something like this. So, now because they are

they are only discrete material points. So, you can number them. So, you can number them using

j. So, j running from 1 to m right. So, j equals 1 to m ok. So, you see everything is not becoming

discrete there is nothing continuum now. So, that's what I say here j. So, j takes the role

of big X 1 big X 2 ok. And you know we you might ask will the 1D crystal really follow this map.

We understand that ok in case of continuum the continuum rod is following this map. If you have

one cross section then you can generate any other cross section using this map. But then when you

have the discrete crystal should it follow the same map right this is continuum this is discrete.

So, that is certainly a question mark isn't it. So, we are just postulating of course that we are

just imposing our continuum map on the discrete nanorod. But you can also prove it if the nanorod

is subjected to let us say pure extension pure torsion or combined extension torsion or pure